When can the components of NEPS of connected bipartite graphs be almost cospectral?

Abstract

We disprove the conjecture of Cvetković posed in 1983 that the components of non-complete extended p-sum (NEPS) of connected bipartite graphs are almost cospectral (cf. D. Cvetković, Publ. Inst. Math. (Beograd) 33 (47) (1983) 29-33). However, we do find the necessary condition for the components to be almost cospectral and we conjecture that it is also the sufficient condition. In the last part of the paper, we are interested in the question when NEPS of bipartite graphs is itself bipartite.